Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

Preface
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students with a thorough understanding of conditional statements, quantifiers, and
logical equivalencies. Emphasis is placed on writing correct and useful negations
of statements, especially those involving quantifiers. The logical equivalencies that
are presented provide the logical basis for some of the standard proof techniques,
such as proof by contrapositive, proof by contradiction, and proof using cases.
The standard methods for mathematical proofs are discussed in detail in Chapter
3. The mathematical content that is introduced to illustrate these proof methods
includes some elementary number theory, including congruence arithmetic. These
concepts are used consistently throughout the text as a way to demonstrate ideas in
direct proof, proof by contrapositive, proof by contradiction, proof using cases, and
proofs using mathematical induction. This gives students a strong introduction to
important mathematical ideas while providing the instructor a consistent reference
point and an example of how mathematical notation can greatly simplify a concept.
The three sections of Chapter 4 are devoted to proofs using mathematical induction.
Again, the emphasis is not only on understanding mathematical induction
but also on developing the ability to construct and write proofs that use mathematical
induction.
The last five chapters are considered “mathematical content” chapters. Concepts
of set theory are introduced in Chapter 5, and the methods of proof studied
in Chapter 3 are used to prove results about sets and operations on sets. The idea
of an “element-chasing proof” is also introduced in Section 5.2.
Chapter 6 provides a thorough study of functions. Functions are studied before
relations in order to begin with the more specific notion with which students
have some familiarity and move toward the more general notion of a relation. The
concept of a function is reviewed but with attention paid to being precise with
terminology and is then extended to the general definition of a function. Various
proof techniques are employed in the study of injections, surjections, composition
of functions, inverses of functions, and functions acting on sets.
Chapter 7 introduces the concepts of relations and equivalence relations. Section
7.4 is included to provide a link between the concept of an equivalence relation
and the number theory that has been discussed throughout the text.
Chapter 8 continues the study of number theory. The highlights include problems
dealing with greatest common divisors, prime numbers, the Fundamental
Theorem of Arithmetic, and linear Diophantine equations.
Finally, Chapter 9 deals with further topics in set theory, focusing on cardinality,
finite sets, countable sets, and uncountable sets.